A Single-Seed Algebraic Identity Linking the Strong and Electroweak Couplings of the Standard Model

We record an exact algebraic identity that relates the strong coupling at the Z pole and the effective leptonic weak mixing angle of the Standard Model. Define the algebraic seed H ≡ 4π + 1 and the two dimensionless quantities A₁ ≡ 5H/8 and A₂ ≡ π/H. Then A₁A₂ = 5π/8 holds as a strict algebraic identity in which H cancels exactly. Identifying A₁ with α_s⁻¹(M_Z) and A₂ with sin²θ_W^eff on the basis of two independent numerical proximities to PDG 2024 central values (within −0.04σ and +0.55σ respectively) yields the prediction α_s⁻¹(M_Z) · sin²θ_W^eff = 5π/8 ≈ 1.96350. The seed H = 4π + 1 admits an elementary geometric reading as the sum of the 0-cell count and the area of the 2-cell in the minimal CW decomposition of the unit 2-sphere, with explicit normalization-dependence caveats. A Monte-Carlo search-space estimate within a deliberately restricted family of 18,496 integer π-polynomial candidates of similar complexity finds three survivors at 10⁻³ precision and none at 10⁻⁴; 5π/8 is the unique single-term member of the survivor list, with the other two requiring explicit integer subtraction. A doubled-box sensitivity check confirms near-linear scaling of survivor counts with box volume. We make no claim of a dynamical derivation and classify the empirical identifications as conjectural numerical proximities. The structurally significant test resides at the factor level: A₂ ↔ sin²θ_W^eff would expand to ∼3.7σ under FCC-ee Z-pole precision, and improved lattice determinations of α_s(M_Z) expected within 5–10 years will sharpen the A₁ ↔ α_s⁻¹(M_Z) test on a comparable timescale.